The curvature is a moment map for the gauge group action on the space of principal connections on a principal bundle over a symplectic manifold. This goes back to [1]}, where this construction is used in the special case of a surface. In the present context this can be realized concretely as follows. Given \(\alpha \in \mathsf {S}^{2}(M)\) , regarded as an element of the Lie algebra \({L}({F})\)  of infinitesimal gauge transformations (see Remark REF ), regard the curvature of \(\nabla \in \mathbb {T}(M, \Omega )\)  as a \({L}({F})\) -valued two-form \(\mathcal {C}\)  and write \(\operatorname{\mathsf {tr}}(\mathcal {C}\alpha ) = R_{ijp}\,^{q}\alpha _{q}\,^{p}\)  for the two-form resulting from tracing the composition of the curvature with the endomorphism \(\alpha _{i}\,^{j}\) . However the dual \({L}({F})^{\ast }\)  is understood, it includes \(\operatorname{Ad}({F})\) -valued \(2n\) -forms.
